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1. Ramanujan’s partition generating functions modulo $$\ell $$

   https://doi.org/10.1007/s11139-025-01241-0  

   Bringmann, Kathrin; Craig, William; Ono, Ken (November 2025, The Ramanujan Journal)

   Abstract For the partition functionp(n), Ramanujan proved the striking identities$$\begin{aligned} \begin{aligned} \mathcal {P}_5(q):=\sum _{n\ge 0} p(5n+4)q^n&=5\prod _{n\ge 1} \frac{\left( q^5;q^5\right) _{\infty }^5}{(q;q)_{\infty }^6},\\ \mathcal {P}_{7}(q):=\sum _{n\ge 0} p(7n+5)q^n&=7\prod _{n\ge 1}\frac{\left( q^7;q^7\right) _{\infty }^3}{(q;q)_{\infty }^4}+49q \prod _{n\ge 1}\frac{\left( q^7;q^7\right) _{\infty }^7}{(q;q)_{\infty }^8}, \end{aligned} \end{aligned}$$ $\begin{array}{c}\begin{array}{cc}{P}_5\left(q\right):=\sum _{n\ge 0}p(5n+4)q^n& =5\prod _{n\ge 1}\frac{{\left(q^5,;,q^5\right)}_{\infty }^5}{{(q;q)}_{\infty }^6},\\ P_7\left(q\right):=\sum _{n\ge 0}p(7n+5)q^n& =7\prod _{n\ge 1}\frac{{\left(q^7,;,q^7\right)}_{\infty }^3}{{(q;q)}_{\infty }^4}+49q\prod _{n\ge 1}\frac{{\left(q^7,;,q^7\right)}_{\infty }^7}{{(q;q)}_{\infty }^8},\end{array}\end{array}$where$$(q;q)_{\infty }:=\prod _{n\ge 1}(1-q^n).$$ ${(q;q)}_{\infty }:={\prod }_{n\ge 1}(1-q^n).$As these identities imply his celebrated congruences modulo 5 and 7, it is natural to seek, for primes$$\ell \ge 5,$$ $\ell \ge 5,$closed form expressions of the power series$$ \mathcal {P}_{\ell }(q):=\sum _{n\ge 0} p(\ell n-\delta _{\ell })q^n\pmod {\ell }, $$ $P_{\ell }\left(q\right):=\sum _{n\ge 0}p(\ell n-{\delta }_{\ell })q^n(mod\ell ),$where$$\delta _{\ell }:=\frac{\ell ^2-1}{24}.$$ ${\delta }_{\ell }:=\frac{{\ell }^2-1}{24}.$In this paper, we prove that$$ \mathcal {P}_{\ell }(q)\equiv c_{\ell } \dfrac{\mathcal {T}_{\ell }(q)}{\left( q^\ell ; q^\ell \right) _\infty } \pmod {\ell }, $$ $P_{\ell }\left(q\right)\equiv c_{\ell }\frac{T_{\ell }\left(q\right)}{{\left(q^{\ell },;,q^{\ell }\right)}_{\infty }}(mod\ell ),$where$$c_{\ell }\in \mathbb Z$$ $c_{\ell }\in Z$is explicit and$${\mathcal {T}}_{\ell }(q)$$ $T_{\ell }\left(q\right)$is the generating function for the Hecke traces of$$\ell $$ $\ell$-ramified values of special Dirichlet series for weight$$\ell -1$$ $\ell -1$cusp forms on$$\textrm{SL}_2(\mathbb Z)$$ ${\text{SL}}_2\left(Z\right)$. This is a new proof of Ramanujan’s congruences modulo 5, 7, and 11, as there are no nontrivial cusp forms of weight 4, 6, and 10. 

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2. Measurement of CP violation in B0 → D+D− and $$ {\textrm{B}}_{\textrm{s}}^0 $$ → $$ {\textrm{D}}_{\textrm{s}}^{+}{\textrm{D}}_{\textrm{s}}^{-} $$ decays

   https://doi.org/10.1007/JHEP01(2025)061  

   Aaij, R; Abdelmotteleb, A_S W; Abellan_Beteta, C; Abudinén, F; Ackernley, T; Adefisoye, A A; Adeva, B; Adinolfi, M; Adlarson, P; Agapopoulou, C; et al (January 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> A time-dependent, flavour-tagged measurement ofCPviolation is performed withB⁰→ D⁺D⁻and$$ {B}_s^0 $$ $B_s^0$→$$ {D}_s^{+}{D}_s^{-} $$ $D_s^{+}{D}_s^{-}$decays, using data collected by the LHCb detector in proton-proton collisions at a centre-of-mass energy of 13 TeV corresponding to an integrated luminosity of 6 fb⁻¹. InB⁰→ D⁺D⁻decays theCP-violation parameters are measured to be$$ {\displaystyle \begin{array}{c}{S}_{D^{+}{D}^{-}}=-0.552\pm 0.100\left(\textrm{stat}\right)\pm 0.010\left(\textrm{syst}\right),\\ {}{C}_{D^{+}{D}^{-}}=0.128\pm 0.103\left(\textrm{stat}\right)\pm 0.010\left(\textrm{syst}\right).\end{array}} $$ $\begin{array}{c}{S}_{D^{+}{D}^{-}}=-0.552\pm 0.100\left(\text{stat}\right)\pm 0.010\left(\text{syst}\right),\\ C_{D^{+}{D}^{-}}=0.128\pm 0.103\left(\text{stat}\right)\pm 0.010\left(\text{syst}\right).\end{array}$ In$$ {B}_s^0 $$ $B_s^0$→$$ {D}_s^{+}{D}_s^{-} $$ $D_s^{+}{D}_s^{-}$decays theCP-violating parameter formulation in terms ofϕ_sand|λ|results in$$ {\displaystyle \begin{array}{c}{\phi}_s=-0.086\pm 0.106\left(\textrm{stat}\right)\pm 0.028\left(\textrm{syst}\right)\textrm{rad},\\ {}\mid {\lambda}_{D_s^{+}{D}_s^{-}}\mid =1.145\pm 0.126\left(\textrm{stat}\right)\pm 0.031\left(\textrm{syst}\right).\end{array}} $$ $\begin{array}{c}{\varphi }_s=-0.086\pm 0.106\left(\text{stat}\right)\pm 0.028\left(\text{syst}\right)\mathrm{rad},\\ \mid {\lambda }_{D_s^{+}{D}_s^{-}}\mid =1.145\pm 0.126\left(\text{stat}\right)\pm 0.031\left(\text{syst}\right).\end{array}$ These results represent the most precise single measurement of theCP-violation parameters in their respective channels. For the first time in a single measurement,CPsymmetry is observed to be violated inB⁰→ D⁺D⁻decays with a significance exceeding six standard deviations. 

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3. Instantaneous everywhere-blowup of parabolic SPDEs

   https://doi.org/10.1007/s00440-024-01263-7  

   Foondun, Mohammud; Khoshnevisan, Davar; Nualart, Eulalia (October 2024, Probability Theory and Related Fields)

   Abstract We consider the following stochastic heat equation$$\begin{aligned} \partial _t u(t,x) = \tfrac{1}{2} \partial ^2_x u(t,x) + b(u(t,x)) + \sigma (u(t,x)) {\dot{W}}(t,x), \end{aligned}$$ $\begin{array}{c}{\partial }_{t}u(t,x)=\frac{1}{2}{\partial }_x^{2}u(t,x)+b\left(u(t,x)\right)+\sigma \left(u(t,x)\right)\dot{W}(t,x),\end{array}$defined for$$(t,x)\in (0,\infty )\times {\mathbb {R}}$$ $(t,x)\in (0,\infty )\times R$, where$${\dot{W}}$$ $\dot{W}$denotes space-time white noise. The function$$\sigma $$ $\sigma$is assumed to be positive, bounded, globally Lipschitz, and bounded uniformly away from the origin, and the functionbis assumed to be positive, locally Lipschitz and nondecreasing. We prove that the Osgood condition$$\begin{aligned} \int _1^\infty \frac{\textrm{d}y}{b(y)}<\infty \end{aligned}$$ $\begin{array}{c}{\int }_1^{\infty }\frac{\text{d}y}{b\left(y\right)}<\infty \end{array}$implies that the solution almost surely blows up everywhere and instantaneously, In other words, the Osgood condition ensures that$$\textrm{P}\{ u(t,x)=\infty \quad \hbox { for all } t>0 \hbox { and } x\in {\mathbb {R}}\}=1.$$ $\text{P}\left\{u\right(t,x)=\infty \text{for all}t>0\text{and}x\in R\}=1.$The main ingredients of the proof involve a hitting-time bound for a class of differential inequalities (Remark 3.3), and the study of the spatial growth of stochastic convolutions using techniques from the Malliavin calculus and the Poincaré inequalities that were developed in Chen et al. (Electron J Probab 26:1–37, 2021, J Funct Anal 282(2):109290, 2022). 

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4. Well-Posedness for Ohkitani Model and Long-Time Existence for Surface Quasi-geostrophic Equations

   https://doi.org/10.1007/s00220-025-05257-x  

   Chae, Dongho; Jeong, In-Jee; Na, Jungkyoung; Oh, Sung-Jin (April 2025, Communications in Mathematical Physics)

   Abstract We consider the Cauchy problem for the logarithmically singular surface quasi-geostrophic (SQG) equation, introduced by Ohkitani,$$\begin{aligned} \begin{aligned} \partial _t \theta - \nabla ^\perp \log (10+(-\Delta )^{\frac{1}{2}})\theta \cdot \nabla \theta = 0, \end{aligned} \end{aligned}$$ $\begin{array}{c}\begin{array}{c}{\partial }_t\theta -{\nabla }^{\perp }log(10+{(-\Delta )}^{\frac{1}{2}})\theta ·\nabla \theta =0,\end{array}\end{array}$and establish local existence and uniqueness of smooth solutions in the scale of Sobolev spaces with exponent decreasing with time. Such a decrease of the Sobolev exponent is necessary, as we have shown in the companion paper (Chae et al. in Illposedness via degenerate dispersion for generalized surface quasi-geostrophic equations with singular velocities,arXiv:2308.02120) that the problem is strongly ill-posed in any fixed Sobolev spaces. The time dependence of the Sobolev exponent can be removed when there is a dissipation term strictly stronger than log. These results improve wellposedness statements by Chae et al. (Comm Pure Appl Math 65(8):1037–1066, 2012). This well-posedness result can be applied to describe the long-time dynamics of the$$\delta $$ $\delta$-SQG equations, defined by$$\begin{aligned} \begin{aligned} \partial _t \theta + \nabla ^\perp (10+(-\Delta )^{\frac{1}{2}})^{-\delta }\theta \cdot \nabla \theta = 0, \end{aligned} \end{aligned}$$ $\begin{array}{c}\begin{array}{c}{\partial }_t\theta +{\nabla }^{\perp }{(10+{(-\Delta )}^{\frac{1}{2}})}^{-\delta }\theta ·\nabla \theta =0,\end{array}\end{array}$for all sufficiently small$$\delta >0$$ $\delta >0$depending on the size of the initial data. For the same range of$$\delta $$ $\delta$, we establish global well-posedness of smooth solutions to the dissipative SQG equations. 

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5. A Weighted Formulation of Refined Decoupling and Inequalities of Mizohata–Takeuchi-Type for the Moment Curve

   https://doi.org/10.1007/s12220-025-02262-3  

   Carbery, Anthony; Li, Zane Kun; Pang, Yixuan; Yung, Po-Lam (February 2026, The Journal of Geometric Analysis)

   Abstract Let$$\Gamma $$ $\Gamma$be a compact patch of a well-curved$$C^{n+1}$$ $C^{n+1}$curve in$$\mathbb {R}^n$$ $R^n$with induced Lebesgue measure$$\textrm{d} \lambda $$ $\text{d}\lambda$, and let$$g \mapsto \widehat{g \,\textrm{d}\lambda }$$ $g\mapsto \widehat{g\text{d}\lambda }$be the Fourier extension operator for$$\Gamma $$ $\Gamma$. Then we have, for arbitrary non-negative weightsw,$$\begin{aligned} \int _{B_R} |\widehat{g \,\textrm{d}\lambda }|^2w \le C_{n,a} R^{a} \sup _S \left( \int _S w\right) \int _\Gamma |g|^2 \, \textrm{d} \lambda \end{aligned}$$ $\begin{array}{c}{\int }_{B_R}|\widehat{g\text{d}\lambda }{|}^{2}w\le C_{n,a}{R}^a\underset{S}{sup}\left({\int }_S,w\right){\int }_{\Gamma }{\left|g\right|}^2\text{d}\lambda \end{array}$for any$$a> \frac{n-3}{2} + \frac{2}{n} - \frac{2}{n^2(n+1)}$$ $a>\frac{n-3}{2}+\frac{2}{n}-\frac{2}{n^2(n+1)}$, where the$$\sup $$ $sup$is over all 1-neighbourhoodsSof hyperplanes whose normals are parallel to the tangent at some point of$$\Gamma $$ $\Gamma$. This represents partial progress on the Mizohata–Takeuchi conjecture for curves in dimensions$$n \ge 3$$ $n\ge 3$, improving upon the exponent$$a=n-1$$ $a=n-1$which can be obtained as a consequence of the Agmon–Hörmander trace inequality. Our main tool in establishing this inequality will be a weighted formulation of refined decoupling for well-curved curves. We also discuss the sharpness of the exponents we obtain in this and in auxiliary results, and further explore this in the context of axiomatic decoupling for curves. 

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